Approximation of ab initio potentials of carbon nanomaterials with machine learning

Sammanfattning: In this work potentials of carbon nanomaterials calculated with Density Functional Theory (DFT) are approximated using an Artificial Neural Network (ANN). Previous work in this field has focused on estimating potential energies of bulk structures. We investigate the possibility to approximate both the potential energies and the forces of periodic carbon nanotubes (CNTs) and fullerenes. The results indicate that for test structures similar to those in the training set the ANN approximates the energies to within 270 meV/atom (< 3.7% error, RMSE 40 meV/atom) and the forces to within 7.5 eV/Å (< 73% error, RMSE 1.34 eV/Å) per atom compared with DFT calculations. Furthermore, we investigate how well the ANN approximates the potentials and forces in structures that are combinations of CNTs and fullerenes (capped CNTs) and find that the ANN generalizes the potential energies to within 100 meV/atom (< 1.1% error, RMSE 78 meV/atom) and the forces to within 6 eV/Å (< 60% error, RMSE 0.55 eV/Å) per atom. The ANN approximated potentials and forces are used to geometry optimize CNTs and we observe that the optimized periodic CNTs match DFT calculated structures and energies while the capped CNTs result in comparable energies but incorrect structures compared to DFT calculations. Considering geometry optimization performed with ANN on CNTs the errors lie within 170 meV/atom (< 1.8% error) with an RMSE of 20 meV/atom. For the geometry optimizations of the capped CNTs the errors are within 430 meV/atom (< 5.5% error) with an RMSE of 14 meV/atom. All results are compared with empirical potentials (ReaxFF) and we find that the ANN approximated potentials are more accurate than the best tested empirical potential. This work shows that machine learning may be used to approximate DFT calculations. However, for further applications our conclusion is that the error of the estimated forces must be reduced further. Finally, we investigate the computing time (number of core hours) required and find that the ANN is about two orders of magnitude faster than DFT and three to four orders of magnitude slower than ReaxFF. For the unseen data the ANN is still around 2 orders of magnitude quicker than the DFT but here it is around 4 order of magnitude slower than ReaxFF.